It remains to verify that \(\varphi (t,x):= [-W(t,x),0]\)  satisfies the condition [i]. Note that by Theorem 2.3, ch 6 [1]}, we have
\(\mathbb {E}\sup _{0\le \tau \le T}\Vert W(\tau ,x)\Vert _{X_\beta } \le \left(\frac{2p}{2p-1}\right)^{2p}\Vert W(T,x)\Vert _{X_\beta } < \infty ,\) 
